On 04/01/18 08:39, Thomas Streicher wrote:
I think what Grothendieck and Verdier could have meant is that Fonc(C,U) doesn't satisfy (C1) and (C2) if C is just a U-category.
For this Ob(C) has to be an element of U as well I guess that's what they call U-petit which is more restrictive than just a U-category.
Thomas
Finally I did a superficial and rapid examination of SGA4. The problem seems to be that for SGA4 U-petit means that it is bijective with an element of U, NOT that it is an element of U. Aparently the universes are not closed in the sense that: 1) X ~ Y, Y belongs U ===> X belongs U which is currently accepted (as we see in Thomas posting above) in the naive practice of category theory with universes. Thus, SGA4 takes a lot of formal trouble to have the representable functors with values in U-sets since hom(x, y) is only bijective with an element of U but is not in general an element of U. In practice of naive category theory with universes we directly assume that hom(x, y) belongs to U (for the U categories) and have the representable functors with no problem, while strictly speaking by definition of U-category, hom(x, y) is U-petit, and U-petit means that it is bijective with an element of U. Thus, if we are naive, U-category means hom(x, y) belongs to U, and take "belongs to U" as the definition "petit". However, Universes are closed in the sense: 2) X subset U, X ~ Y, Y belongs U ===> X belongs U (this is true because X is the union of its singletons which are in U) Which is relevant in Paul question. SGA4 C1) and C2) are equivalent to what Paul calls "U-included" a) C subset U b) hom(x, y) belongs U Now, if a) holds, then if hom(x, y) is U-petit then it belongs to U ((*) can you proved this using 2) ?). Thus U-categories in which a) holds satisfy automatically b), thus are "U-included" in a rigorous non naive sense (provided you can prove (*)) Detailed proofs in the style of SGA4 I 1.3 or Paul's proof in this thread are necessary to resolve seriously this question. Can you justify (or prove, or explain) SGA4 statement: "Soit C une categorie appartenant a U . Alors la categorie Fonct(C, D) ne possede pas en general les proprietes (C1) et (C2). Par exemple la categorie Fonct(C,U-Ens) ne possede aucune des proprietes (C1) et (C2)." On the other hand, if you practice naive category theory with universes then all problems dissapear :=) Eduardo. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]